Radian Measure of Angles - Problem 3

Norm Prokup

We're talking about radian measure of angles. I have a problem here that asks me to express one degree in radians and one radian in degrees. This is a useful problem because as a result of it we'll get some conversion factors that will make it really easy to switch back and forth.

One degree in radians, always remember that 180 degree equals pi. 180 degrees equals pi radians, so to get one degree divide both sides by 180. One degree is pi over 180 and if you want a decimal value for this you can use your calculator pi divided by 180, this is approximately .0175.

Okay convert one radian into degrees, one radian no let me start with pi radians equals 180 degrees. If I divide both sides by pi, I get one radian equals 180 degrees over pi and that would be the reciprocal of what we had before, it's about 57.3 degrees, so a little less than 60 degrees is one radian.

This leads to the conversion factors. One way to convert from degrees to radians is take the number of degrees and multiply it by this conversion factor, pi over 180, multiplied by pi over 180 and you'll get your radians. To get from radians to degrees, you multiply by this conversion factor times 180 over pi and we'll use those conversion factors in a second.

One thing I want to talk about here is in advanced Math, we tend to think of angles as rotations of a ray about a point, so when you look at the initial side and the terminal side of an angle, imagine the initial side has rotated around to the terminal side and that kind of thinking allows us to envision angles like 390 degrees which is greater than one revolution. It also allows us to think about negative angles like this one -3 pi over 4, the initial side has rotated clockwise 3 pi over 4.

In Mathematics, the clockwise direction is defined as negative, counter clockwise is positive, that's really important. Now you will notice that in both of these cases I drew the angles superimposed on the coordinate system with the vertex right at the origin and the initial side on the positive x axis, this is called standard position and this is the way we're going to be drawing angles for the rest of the course, but you'll see this a lot in any Pre-calculus course. These are angles drawn in standard position.

Let's convert each of these to the other form of measure, this one is in degrees, so I want to convert it into radians. I can use my conversion factor here 390 degrees times pi over 180. Now there's a lot of common factors here, this is 30 times 6, this is 30 times 13 and the 30s can cancel and you're left with 13 pi over 6.

In this case we've got -3 pi over 4, we want to convert these to degrees, so -3 pi over 4. We want to multiply by 180 over pi, 180 degrees over pi. Remember in the conversion factor if you have a hard time remembering which is which, when the degrees are in the numerator, your answer is going to be in degree. When your degrees are in the denominator, your answers are going to be in radians.

So here I just want to cancel the pi's, 4 cancels with 180 leaving 45 and -3 times 45 is -135 degrees. So you don't have to use the conversion factors to convert from degrees to radians, you could always use this fact, but the conversion factors are handy for people who like formulas. And also don't forget, the measure of angle in standard position, this is the way we'll draw angles from now on.